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© 2002 by Shawna Haider
There are two kinds of notation for graphs of
inequalities: open/filled-in circle notation and
interval notation brackets.
x  1
[
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
circle filled in
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
squared end bracket
Because the inequality is "greater than
or equal to" the solution includes the
endpoint. That’s why the circle is filled in.
With interval notation brackets, a square
bracket means it includes the endpoint.
Let's look at the two different notations with a
different inequality sign.
x  1
)
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
circle not filled in
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
rounded end bracket
Since this says "less than" we make the arrow go the
other way. Since it doesn't say "or equal to" the
solution cannot equal the endpoint. That’s why the
circle is not filled in. With interval notation brackets, a
rounded bracket means it cannot equal the endpoint.
Graphically there are two different notations, and
when you write your answers you can use
inequality notation or interval notation. You
should be familiar with both.
[
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
x  1
Inequality notation
for graphs shown
above.
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
[1, )
Interval notation
for graphs shown
above.
Let's have a look at the interval notation.
[1, )
[
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
For interval notation you list the left endpoint, a
comma, and then the right endpoint – so the
solutions are any numbers that are between the
left and right endpoints.
The bracket after the infinity sign is rounded because infinity
is not a real number and so the solution does not include the
endpoint (there is no endpoint).
Let's try another one.
(2,4]
Rounded bracket means
cannot equal -2
Squared bracket means
can equal 4
The brackets used in the interval notation above
are the same ones used when you graph this.
(
]
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
This means everything between –2 and 4 but not including -2
Let's look at another one
(,4)
)
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
Compound Inequalities
Let's consider a "double inequality"
(having two inequality signs).
2 x3
(
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
]
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
This is a compound inequality because it is a
combination of two inequalities. It reads that “x is
greater than –2 and x is less than or equal to 3”.
The word "and" means both parts must be true.
Compound Inequalities
Now let's look at another form of a "double inequality"
(having two inequality signs).
x  2 or x  3
)
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
[
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
Instead of "and", these are "or" problems. One part
or the other part must be true (but not necessarily
both). Either “x is less than –2 or x is greater than
or equal to 3”. (In this case both parts cannot be
true at the same time since a number can't be less
than –2 and also greater than 3).
Now let's look at an "or" compound inequality
x  2 or x  3
)
[
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
(,2)
 [3, )
When the solution consists of more than one interval,
we join them with a union sign.
There are two intervals to list when you list in
interval notation.
Solving Linear Inequalities.
Essentially, everything you did to solve linear
equations applies to solving linear inequalities
except that if you multiply or divide by a
negative you must reverse the inequality sign.
So to solve an inequality just do the same steps as
with an equality to isolate the variable, but WATCH
OUT WHEN YOU MULTIPLY OR DIVIDE BY A
NEGATIVE NUMBER.
Example:
2x  6  4x  8
- 4x
- 4x
 2x  6  8
+ 6 +6
 2 x  14
-2
We flipped the sign!
-2
x  7